Small Perturbations of Stochastic Differential Equations with Power Coefficients

Юрій Євгенович Приходько

Abstract

Background. Random perturbations of ordinary differential equations were considered by Bafico (1980), Bafico, Bal­di (1982), Delarue, Flandoli (2014), Delarue, Flandoli, Vincenzi (2014), Krykun, Makhno (2013), Pilipenko, Pro­ske (2015). Bafico, Baldi (1982) considered random perturbation of the differential equation that describes the Peano phenomenon. The coefficients of the initial differential equation are not Lipschitz continuous, so there may be no uniqueness of the solution. Then stochastic differential equation is considered instead of ordinary differential equation and the weak convergence of its solutions is proved.

Objective. The aim of this paper is to generalize the result of Bafico, Baldi (1982) to the case of stochastic differential equation dX(t)=a(X(t))dt+σ(X(t))dW(t) with power coefficients.

Methods. Small random perturbations of the initial equation dX(t)=a(X(t))dt+(ε+σ(X(t)))dW(t) are considered and the limit behaviour of its solutions is studied.

The methods used to prove the weak convergence of the solutions are based on the methods developed in Pilipenko, Prykhodko (2015 and 2016).

Results. The limit behaviour of the solutions of stochastic differential equations with perturbations is considered and the weak convergence of such solutions is proved.

Conclusions. The result of Bafico, Baldi (1982) is thus generalized to the case of stochastic differential equation with power coefficients.